Question

a) Berechnen Sie das Volumen eines Zylinderringes mit den Radien \(R_{1}\) und \(R_{2}\). b) Bestimmen Sie den Flächeninhalt eines Halbkreises mit Hilfe eines Zweifachintegrals.

Short answer

a) The volume of a cylindrical ring with radii \(R_{1}\) and \(R_{2}\) is \(\pi \cdot h \cdot (R_{2}^{2} - R_{1}^{2})\), where h is the height of the cylinders. b) The area of a semi-circle can also be found using a double integral, which gives the same result as the formula \(\frac{1}{2} \pi \cdot r^{2}\), where r is the radius of the semicircle.

Step-by-step solution

Part A - Step 1: Identify the formula for volume of a cylindrical ring.

To find the volume of a cylindrical ring, we need to find the difference in the volumes of two cylinders. The volume of a cylinder is given by the formula \(\pi \cdot r^{2} \cdot h\), where r is the radius and h is the height.

Part A - Step 2: Apply the formula.

In our case, we have two radii, \(R_{1}\) and \(R_{2}\). If we assume that both cylinders have the same height, say h, the volume of the cylindrical ring (V) will be \(\pi \cdot h \cdot (R_{2}^{2} - R_{1}^{2})\).

Part B - Step 1: Define a semi-circle.

A semi-circle is half a circle. Its area is defined by \(\frac{1}{2} \cdot \pi \cdot r^{2}\), where r is the radius. To conduct the double integral, we first need to parameterize the semi-circle.

Part B - Step 2: Parameterize the semi-circle.

We use polar coordinates to parameterize the semi-circle, with \(r\) ranging from 0 to \(R\) and \(\theta\) ranging from 0 to \(\pi\), where \(R\) is the radius of the half circle. Then the coordinates \((x, y)\) of the points in the semi-circle are given by \((r \cos \theta, r \sin \theta)\).

Part B - Step 3: Conduct the double integral.

Using the double integral, the area (A) of the semi-circle can be found as \(A = \int_{0}^{\pi} \int_{0}^{R} r dr d\theta\). Evaluating the integral gives \(A = \frac{1}{2} \pi R^{2}\), which confirms our earlier definition of the area of a semi-circle.

Key concepts

Integrals in Physics

Integrals are a fundamental part of physics, providing a mathematical means to calculate quantities that involve accumulation or distribution. For example, when we want to calculate the volume of objects like cylindrical rings, integrals allow us to sum up all the tiny volumes that make up the whole object. In the context of our exercise, integrals are used to differentiate between the volumes of two cylinders to find the volume of a cylindrical ring.

Moreover, integrals in physics are not confined to calculating volumes. They are pivotal in various other areas, such as determining the area under a curve, finding the center of mass, or in our case, calculating the area of a semi-circle using a double integral. Double integrals extend the concept of integration to two dimensions, enabling the calculation of areas and volumes in more complex shapes.

Volume Calculation

When it comes to volume calculation, understanding the shape of the object is crucial. For a cylindrical ring, we consider the object as the difference between two cylinders. Here's where the mathematical simplicity and beauty of integration shine. Utilizing the formula for the volume of a cylinder, \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height, we can simply subtract the smaller cylinder's volume from the larger one's.

Applying this knowledge to the cylindrical ring with radii \( R_1 \) and \( R_2 \) and a common height \( h \) helps us compute the volume \( V = \pi h (R_2^2 - R_1^2) \). Clear illustrations and step-by-step solutions greatly aid in grasping this concept and avoiding common mistakes like mixing up radii or forgetting the height component.

Polar Coordinates

Polar coordinates are an alternative to the traditional Cartesian coordinate system. Instead of using x and y axes, polar coordinates define a point based on its distance from a reference point (the pole) and the angle from a reference direction. This system is incredibly useful when dealing with problems that have circular or spherical symmetry, like the semi-circle in our exercise.

Parameterizing a semi-circle using polar coordinates involves setting a radius range from 0 to \( R \) and an angle \( \theta \) range from 0 to \( \pi \) radians. Each point within the semi-circle can then be described using \( (r \cos \theta, r \sin \theta) \) coordinates, where \( r \) is the distance from the origin, and \( \theta \) is the angle. This parameterization is key when setting up integrals for area and volume, as it simplifies the integration process and leads to more straightforward solutions.